# Probabilistic Palmgren-Miner Rule for Solder Materials Experiencing Elastic Deformations

## Background/Incentive

It has been recently shown that there are effective ways not only to reduce the interfacial stresses in electronic packaging assemblies with solder joint arrays as the second level of interconnections, but to do that to an extent that inelastic strains in the peripheral joints, where the induced thermal stresses and strains are the highest, are avoided. While various and numerous modifications of the empirical Coffin-Manson relationship are used to predict the fatigue life of solder materials experiencing inelastic strains and operated in low-cycle fatigue conditions, the Palmgren-Miner rule of the linear accumulation of fatigue damages, although suggested many decades ago, is still viewed by many material scientists and reliability physicists as a suitable model that enables one to quantify the cumulative fatigue damage in metals experiencing elastic strains.

In this analysis, the Palmgren-Miner rule is extended for the case of random loading, and a simple formalism is suggested for the evaluation of the remaining useful lifetime (RUL) for a solder material subjected to random loading and experiencing elastic thermally induced shearing deformations. Special highly focused and highly cost-effective accelerated tests have to be conducted, of course, to establish the S-N curve for the given solder material. In future work, we intend to extend the suggested methodology to take into account various aspects of the physics-of-failure: the role of the growth kinetics of intermetalic compound (IMC) layers; the random number, size and orientation of grains in the joints; position of the joint with respect to the mid-cross section of the assembly (peripheral joints are more prone to elevated interfacial stresses); and assembly size. However, all this effort, important as it is, is beyond the scope of this analysis, which is aimed at the extension of the classical Palmgren-Miner rule for the case of random loading.

Solder joints employed in today’s IC packaging engineering as second-level interconnections provide both electrical connection and mechanical support [12-14]. In the latter capacity, they are subjected to thermally induced stresses primarily caused by the (global) thermal expansion (contraction) mismatch of the dissimilar materials of the package and the substrate. The induced interfacial stresses concentrate at the assembly ends and could be quite high [15-17], thereby compromising the integrity of the peripheral joints. Therefore, there is an obvious incentive to minimize the induced stresses for the improved short- and long-term reliability of the package design.

It has been recently shown [18-26] that effective ways exist not only to reduce the interfacial stresses in packaging assemblies with solder joint interconnections, but to do that to an extent that inelastic strains in the peripheral joints, where the induced stresses and strains are the highest, are avoided [27-34]. This could be achieved, particularly, by using solder joints with elevated stand-offs and/or by employing inhomogeneous solder joint systems, in which the peripheral zones of the package are characterized by lower Young’s moduli and/or lower soldering temperatures. If such an effort is successful, the high thermal stresses in the solder material, when the assembly is fabricated at the elevated temperature and is subsequently cooled down to a low (room or testing) temperature, and/or when the assembly experiences temperature cycling during reliability testing and actual operation, will be replaced by the temperature cycling within the elastic range.

It goes without saying that such a situation would be highly desirable and result in a considerably longer lifetime of the solder material. As is known (see, for example, [35-38]), the reliability of a material has to be checked and assured with respect to two major loading conditions: short-term high-level loading (ultimate strength) and long-term relatively-low-level, but repetitive, loading (fatigue strength). The ultimate strength (short-term reliability) is defined by the ability/capacity of the material to withstand a significant load applied just once. This strength is measured by the maximum (“ultimate”) stress that a material is able to withstand in a typical loading condition (usually in tension or in shear) before breaking. Shear-off product development testing is a typical example of such a condition. As to the fatigue/endur- ance strength (long-term reliability) of an elastic material, it is defined by the highest stress that this material can withstand for the given number of cycles without breaking. Fatigue failure occurs because of the accumulation of micro-damages that result in the developing and propagation of fatigue cracks. In electronics reliability, the fatigue strength (lifetime, number-of-cycles-till-failure) is usually determined by temperature cycling.

The research literature in this field is enormous, especially when addressing situations, when the solder joints experience inelastic deformations. Since it has been shown that in many situations such deformations could be avoided by taking appropriate design measures, in the analysis that follows, a methodology for the evaluation of the remaining useful lifetime (RUL) for a solder material experiencing elastic thermally induced shearing deformations is addressed.

It is assumed that this could be done, in the first approximation and for the preliminary and tentative evaluations, on the basis of the simplest and well-known Palmgren-Miner rule of linear accumulation of fatigue damages [28, 29] and that this rule could be extended for the situation when thermal loading is applied in a random fashion, which is a typical situation in electronics and photonics systems operation. Certainly, special accelerated tests are required to establish the S-N curve for the given solder material in a situation when its yield stress is not exceeded. This should be done with consideration of the degradation effects of the solder material and its interfaces (see, for example, [41-48]).

## Probabilistic Palmgren-Miner Rule

A typical stress versus number-of-cycles-to-failure (Wohler) S-N curve/diagram in logarithmic coordinates is shown in Figure 4.7 for the case when a variable tensile stress is applied. When the applied stress is in shear, this diagram can be approximated by a power law (see Figure 4.8):

Here, *N _{f}* is the number of cycles corresponding to reaching the fatigue curve,

*x*is the level of the steady-state fatigue,

_{f }*x„*is the amplitude of the variable shearing stress, and

*m =*tana is the tangent of the angle that the limited-fatigue portion of the diagram forms with the vertical line that divides the limited fatigue and the steady- state fatigue regions.

In accordance with the Palmgren-Miner theory of the linear accumulation of fatigue damages, these damages are independent of the degree of the consumption of the fatigue lifetime at the given moment of time. The accumulated damages are also independent of the “prehistory” of loading, and therefore those that are caused by the current loading cycle can simply be added to previous damages. Such an assumption seems to be particularly justified in the case of random loading, when,

FIGURE 4.7 Yield-stress to maximum-elastic-stress ratios versus product of the parameter of the interfacial shearing stress and half-assembly length for different ratios of the length of the inelastic zone to half-assembly length.

because of the sequential action of cycles with high and low stresses, the material’s weakenings caused by high-stress cycles interchange with its strengthenings caused by low-stress cycles, so that the effect of the “prehistory” of loading is being continuously smoothed down. The accumulated fatigue damage is assessed as

FIGURE 4.8 Stress versus number-of-cycles-to-failure (Wohler) curve/diagram in logarithmic coordinates.

*п*

Here, = — is the damage from the /th level of loading, *к* is the total number of the

loading level, и, is the number of loading cycles of the /'th level, and *N,* is the number of loading cycles of the /th level leading to failure. Such a failure can be caused either by a single loading of the /th level, or by the entire spectrum of loadings of different levels. Assuming that the work *W* of the external loading leading to fatigue failure is the same in both cases, one has

where *W,* is the work of «, cycles of a variable loading of the /th level. This relationship yields

By summing up the elementary works *W,* in both parts of this equation, the following Palmgren-Miner formula for the linear accumulation of fatigue damages can be obtained:

When the material experiences continuous random loading, this formula can be generalized as follows:

Here, *N* is the number of cycles that corresponds to achieving the fatigue curve, and *N _{f}* is the number of cycles till fatigue failure.

## Remaining Useful Life

The total number of cycles accumulated for the time *t = RUL* can be naturally determined as *N, = —*, where *t _{e}* is the effective period of random loading. Assuming that

the loading cycles are distributed in a uniform fashion during the material’s lifetime *t = RUL* within the interval 0 ч *N* >- *N,*, the probability density function for the number *N* of cycles is

Using the condition

of the equality of the elementary probabilities for the random amplitudes x_{a} of loading and the number *N* of cycles we have

In formulas (4.62) and (4.63), /_{T}(x„) is the probability density distribution function for the random amplitudes x„ of loading. Introducing formulas (4.55) and (4.63) into condition (4.60), the following expression for the time-to-failure (RUL) can be obtained:

where the factor

considers the roles of the material properties (through the ultimate shearing stress x„) and the level of loading (through the level and distribution of random amplitudes of the shearing stress). The ultimate shearing stress t„ could be accepted in this analysis, limited to the elastic deformations, equal to the yield stress *x _{Y}* of the material.

Two important aspects of the fatigue limit *x _{f}* should be pointed out.

The fatigue limit *x _{f},* even when it is low, as it is in the case in question, could still decrease, because of the material aging/degradation, with an increase in the number of cycles. In this connection, we would like to indicate that there is a way to separate the irreversible and unfavorable physics-of-failure related degradation process, which results in the increase in the failure rate with time and in a lower fatigue limit, from the also irreversible, but favorable, statistics-related process that results in the decreased failure rate with time [36, 37]. This aspect of the assessment of the fatigue lifetime is, however, beyond the scope of this analysis. Another aspect has to do with the fact that the fatigue limit

*x*depends on the loading spectrum: the larger the portion of this spectrum above the fatigue limit, the larger the rate of the decrease in this limit. In an approximate analysis, one could assume that the rate in the decrease in the fatigue limit depends on its initial value x°, the rate

_{f}*^*of the

accumulation of fatigue damages and the total level *N* of the accumulated damages at the given moment of time. These considerations could be formalized in the following equation:

where *x _{f}* is the fatigue limit after

*n*loadings, t" is the initial value of the fatigue limit,

*N*is the number of cycles till fatigue failure, and a is the materials parameter. Equation (4.66) has the following solution:

As evident from this solution, the fatigue limit *x _{f}* depends, for the given mate-

*n*

rial, characterized by the exponent a, on the accumulated fatigue damage —, and

becomes zero at the moment of failure. The role of the variability of the fatigue limit is, however, beyond the scope of this analysis.

## Rayleigh Law for the Random Amplitude of the Interfacial Shearing Stress

The long-term distributions of the stress amplitudes could be exponential, or distributed in accordance with the log-normal law, or with the Weibull law. Let us address, as a suitable example, the case when the random amplitude of the interfacial shearing stress is distributed in accordance with Rayleigh law

Here *D _{x}* is the variance of the random amplitude

*x„*and x* = ^/d7 is the most likely value (mode) of this amplitude. Introducing expression (4.68) into formula (11), we obtain

When the loading cycle is asymmetric, the random amplitudes *x _{a}* should be multiplied by the factor

*к*» that can be determined from Goodman’s law (see, for example, [49]) as

Here,

is the mean value of the stress cycle. Then, formula (4.69) yields where the following notations are used:

The integral in formula (4.72) can be taken numerically, but could also be expressed through tabulated functions. Since the integrand in this integral does not make physical sense for the shearing stress amplitudes exceeding the yield stress, one could put the upper limit in this integral equal to infinity. Then,

where

is the gamma function and

is the Pierson function. Both functions are tabulated (see, for example, [37]). The coefficient in front of the Pierson function in (4.74) can be computed as

for odd *m* numbers and as
for even m numbers.

## Numerical Example

Input data

Structural Element |
Package (Comp. J+1) |
PCB (Comp. #2) |
Solder (96.5%Ag3.5%Sn) |

Effective Young’s modulus, kg/mm |
8775.5 |
2321.4 |
1939.0 |

Poisson’s ratio |
0.30 |
0.30 |
0.38 |

CTE, l/°C |
6.5 xIO' |
15.0x10-" |
X |

Thickness, mm |
2.0 |
1.5 |
0.2 |

Estimated yield stress of the solder material in shear *x _{Y} =* 1.700 kg/mm

^{2};

Soldering temperature 160°C; Lowest testing temperature-20°C; Highest testing temperature: +100°C;

Largest external thermal strain:

e_{max} = ДаД/,_{шх} = (15.0 - 6.5)1 O'^{6} x 180 = 153.0 x 10'^{5};

Smallest external thermal strain:

e_{min} = ДаД/_{т|}„ = (15.0-6.5)10" x60 = 51.0x 10~^{5};

The fatigue limit is *x _{f}* = 0.200 kg/mm

^{2};

Most likely interfacial shearing stress in actual operation condition is t = 1.00 kg/mm^{2};

Factor considering the slope of the limited fatigue region: m = 10;

Effective period of random loading: *t _{e}* = 24 hours;

Number of cycles to failure *N _{f}* = 10

^{4}.

Calculated data

Axial compliances of the assembly components: Flexural rigidities of the assembly components:

Axial compliance of the assembly:

Shear moduli:

• of the package

• of the PCB

• of the solder

Interfacial compliances:

Parameter of the interfacial shearing stress

Highest interfacial shearing stress Lowest interfacial shearing stress Mean value of the stress cycle

Factor accounting for the ratio of the fatigue limit for the asymmetric cycle to the fatigue limit for the symmetric cycle

The lower limit in the integral (4.74)

Pierson function [49]

Then, formula (4.72) yields

We conclude that

- • A simple and practically useful methodology is suggested for the evaluation of the RUL for a solder material experiencing elastic thermally induced shearing deformations.
- • The classical Palmgren-Miner rule is extended for the case of random loading. Certainly, special accelerated tests are required to establish the S-N curve for the given solder material in a situation when its yield stress is not exceeded.
- • In future work, we intend to extend the suggested methodology to take into account various aspects of the physics-of-failure: the role of the growth kinetics of IMC layers; the random number, size, and orientation of grains in the joints; position of the joint with respect to the mid-cross section of the assembly (peripheral joints are more prone to elevated interfacial stresses); and assembly size.

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